MATRIISILASKENNAN TEHTÄVIÄ

Ominaisarvot ja ominaisvektorit

Edellinen Hakemisto

Laske seuraavien matriisien ominaisarvot ja ominaisvektorit.

Tehtävä

Ratkaisu

>> A = [1 0 ; 0 -4 ]

>> A = [1 2 ; 0 3 ]

Rakaisu
>> A = [3 4; 4 -3]

A =

3 4
4 -3

>> eig(A)

ans =

-5
5

>> [V,D] = eig(A)

V =

0.4472 -0.8944
-0.8944 -0.4472

D =

-5 0
0 5

Ratkaisu

>> A = [1 0; 0 1]

A =

1 0
0 1

>> eig(A)

ans =

1
1

>> [V,D] = eig(A)

V =

1 0
0 1

D =

1 0
0 1

Ratkaisu

>> A = [0 1; 1 0]

A =

0 1
1 0

>> eig(A)

ans =

-1
1

>> [V,D] = eig(A)

V =

-0.7071 0.7071
0.7071 0.7071

D =

-1 0
0 1

Ratkaisu

>> A = [-5 2; -9 6]

A =

-5 2
-9 6

>> eig(A)

ans =

-3
4

>> [V,D] = eig(A)

V =

-0.7071 -0.2169
-0.7071 -0.9762

D =

-3 0
0 4

Ratkaisu

>> A = [0 1 ; -1 0]

A =

0 1
-1 0

>> eig(A)

ans =

0 + 1.0000i
0 - 1.0000i

>> [V,D] = eig(A)

V =

0.7071 0.7071
0 + 0.7071i 0 - 0.7071i

D =

0 + 1.0000i 0
0 0 - 1.0000i

Ratkaisu

>> A = [0.8 -0.6 ; 0.6 0.8]

A =

0.8000 -0.6000
0.6000 0.8000

>> eig(A)

ans =

0.8000 + 0.6000i
0.8000 - 0.6000i

>> [V,D] = eig(A)

V =

0 - 0.7071i 0 + 0.7071i
-0.7071 -0.7071

D =

0.8000 + 0.6000i 0
0 0.8000 - 0.6000i

Ratkaisu

>> A = [3 0 0;5 4 0; 3 6 1]

A =

     3     0     0
     5     4     0
     3     6     1

>> eig(A)

ans =

     1
     4
     3

>> [V,D] = eig(A)

V =

         0       0            0.0693
         0       0.4472   -0.3465
    1.0000    0.8944   -0.9355

D =

     1     0     0
     0     4     0
     0     0     3

Ratkaisu

>> A = [1 2 -3;-2 -1 6;1 2 0]

A =

     1     2    -3
    -2    -1     6
     1     2     0

>> eig(A)

ans =

-1.5000 + 0.8660i
-1.5000 - 0.8660i
3.0000

>> [V,D] = eig(A)

V =

   0.7638                    0.7638               -0.0000
-0.5455 + 0.1890i   -0.5455 - 0.1890i   0.8321
   0.2728 - 0.0945i     0.2728 + 0.0945i   0.5547

D =

-1.5000 + 0.8660i        0                              0
        0                        -1.5000 - 0.8660i        0
        0                         0                              3.0000

Ratkaisu

>> A = [1 2 -3; 2 4 -6; -1 -2 3]

A =

     1     2    -3
     2     4    -6
    -1    -2     3

>> eig(A)

ans =

         0
    8.0000
         0

>> [V,D] = eig(A)

V =

   -0.9526    0.4082    0.9526
    0.2722    0.8165   -0.2722
   -0.1361   -0.4082    0.1361

D =

         0         0           0
         0      8.0000      0
         0         0           0

Edellinen Hakemisto